L(s) = 1 | − 2-s − 1.46·3-s + 4-s + 2.24·5-s + 1.46·6-s − 0.440·7-s − 8-s − 0.865·9-s − 2.24·10-s − 1.12·11-s − 1.46·12-s − 4.90·13-s + 0.440·14-s − 3.28·15-s + 16-s + 4.96·17-s + 0.865·18-s − 6.89·19-s + 2.24·20-s + 0.643·21-s + 1.12·22-s + 23-s + 1.46·24-s + 0.0611·25-s + 4.90·26-s + 5.64·27-s − 0.440·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.843·3-s + 0.5·4-s + 1.00·5-s + 0.596·6-s − 0.166·7-s − 0.353·8-s − 0.288·9-s − 0.711·10-s − 0.339·11-s − 0.421·12-s − 1.35·13-s + 0.117·14-s − 0.848·15-s + 0.250·16-s + 1.20·17-s + 0.203·18-s − 1.58·19-s + 0.503·20-s + 0.140·21-s + 0.239·22-s + 0.208·23-s + 0.298·24-s + 0.0122·25-s + 0.961·26-s + 1.08·27-s − 0.0832·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6447740739\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6447740739\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 + 1.46T + 3T^{2} \) |
| 5 | \( 1 - 2.24T + 5T^{2} \) |
| 7 | \( 1 + 0.440T + 7T^{2} \) |
| 11 | \( 1 + 1.12T + 11T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 19 | \( 1 + 6.89T + 19T^{2} \) |
| 29 | \( 1 + 0.760T + 29T^{2} \) |
| 31 | \( 1 + 8.34T + 31T^{2} \) |
| 37 | \( 1 + 7.03T + 37T^{2} \) |
| 41 | \( 1 - 4.52T + 41T^{2} \) |
| 43 | \( 1 - 6.09T + 43T^{2} \) |
| 47 | \( 1 - 6.01T + 47T^{2} \) |
| 53 | \( 1 + 6.57T + 53T^{2} \) |
| 59 | \( 1 + 9.59T + 59T^{2} \) |
| 61 | \( 1 + 6.70T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 9.48T + 73T^{2} \) |
| 79 | \( 1 - 1.45T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 17.9T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.979032586885782651450749734401, −7.42082938546852584625275454582, −6.54648050127370302870739712793, −6.00210911484928664761513450798, −5.38954230431729838362162760321, −4.78766944432210782315274927957, −3.47082691548522229603311565208, −2.46068327672079945400058337560, −1.82668802647234219437901170882, −0.46855857139059959397666614695,
0.46855857139059959397666614695, 1.82668802647234219437901170882, 2.46068327672079945400058337560, 3.47082691548522229603311565208, 4.78766944432210782315274927957, 5.38954230431729838362162760321, 6.00210911484928664761513450798, 6.54648050127370302870739712793, 7.42082938546852584625275454582, 7.979032586885782651450749734401